Lemma 2.1 ([1]}) Let
\(\mathcal {Z}\in {\rm Mat}_{n,k+m}^{>0}\) . Write \(Y\in {\rm Gr}_{k,k+m}\)  as \(Y=C\mathcal {Z}\) 
with \(C\in {\rm Gr}_{k,n}\) . We have
\(\left\langle C\mathcal {Z},i_{1},\dots ,i_{m}\right\rangle =\sum _{J=\left(j_{1}<\cdots <j_{k}\right)\in \binom{\left[n\right]}{k}}p_{J}\left(C\right)\left\langle j_{1},\dots ,j_{k},i_{1},\dots ,i_{m}\right\rangle ,\) 
